https://file.aiquickdraw.com/mathbot/file/17289317721475ea8tn3x.jpg

The problem in the image asks to calculate:

1. **Percentile for 43**: This is to determine the percentile rank for the score of 43.
2. **Value corresponding to the 68th percentile (\(P_{68}\))**: This involves finding the value that corresponds to the 68th percentile.

Let's solve each part.

---

### (a) Find the percentile for 43

The scores are:  
\[
43, 45, 46, 49, 51, 53, 59, 62, 67, 68, 71, 74, 78, 81, 83, 86, 88, 91, 95, 97
\]

To find the percentile for 43:
- **Step 1**: Arrange the scores in ascending order (already done here).
- **Step 2**: Use the formula to calculate the percentile rank:

\[
\text{Percentile} = \frac{\text{Number of values below } 43}{\text{Total number of values}} \times 100
\]

In the list of scores, no value is below 43 (it is the smallest). Therefore, the percentile rank for 43 is:

\[
\text{Percentile} = \frac{0}{20} \times 100 = 0\%
\]

So, the percentile for 43 is **0%**.

---

### (b) Find \(P_{68}\)

To find the value corresponding to the 68th percentile, we use the formula:

\[
P_k = \left( \frac{k}{100} \times (n+1) \right)
\]
where \(k = 68\) and \(n = 20\).

\[
P_{68} = \left( \frac{68}{100} \times (20 + 1) \right) = 14.28
\]

Since the result is not a whole number, we interpolate between the 14th and 15th values. The 14th value is 81, and the 15th value is 83. To interpolate:

\[
P_{68} = 81 + (83 - 81) \times 0.28 = 81 + 0.56 = 81.56
\]

So, the value that corresponds to the 68th percentile is approximately **81.56**.

---

Would you like further details on any of these steps or have any additional questions? 

Here are some related questions:
1. How do you find the median from a set of values?
2. What is the formula for calculating quartiles?
3. How do you calculate the interquartile range (IQR)?
4. Can we use the same method to calculate any percentile?
5. What does a higher percentile indicate about a score?

**Tip:** Percentiles are useful for understanding how a specific score compares to others in a data set.

A statistics professor randomly selects 20 of her students and records their final exam scores. (a) Find the percentile for 43. (b) Find P68 (the value that corresponds to the 68th percentile).

Learn how to find the percentile rank for a score of 43 and determine the 68th percentile value from a data set of 20 final exam scores. This problem covers key concepts in statistics, including percentile calculations and data analysis. Suitable for students in Grades 10-12.

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